… We prove that the chromatic polynomial PG(q) of a finite graph G of maximal degree ∆ is
free of zeros for |q| 李 C∗(∆) with … For any integer q, let PG(q) be equal to the number of
proper colourings with q colours of the graph G, that is, colourings such that no two adjacent
vertices of the graph have equal colours. The function PG(q) is a polynomial known as the
chromatic polynomial, and it coincides with the partition function of the anti-ferromagnetic
Potts model with q states on G at zero temperature. Sokal [12] exploited a well-known …

We prove that the chromatic polynomial $ P_\mathbb {G}(q) $ of a finite graph $\mathbb {G}
$ of maximal degree? is free of zeros for| q|? C*(?) with $$ C^*(\D)=\min_ {0< x< 2^{\frac
{1}{\D}}-1}\,\frac {(1+ x)^{\D-1}}{x\,[2-(1+ x)^\D]}. $$ This improves results by Sokal and Borgs.
Furthermore, we present a strengthening of this condition for graphs with no triangle-free
vertices.